Question

Consider the function f(x) = x^3 + x^2 - x + 9 on the interval [-2, 0]. Find f'(x) and the critical value of f(x) on the interval [-2, 0].

Answer 1

To find f'(x) and the critical value of f(x) on the interval [-2, 0], use the power rule to differentiate f(x) and solve for x where f'(x) = 0.

Step-by-step explanation:

To find the derivative of the function f(x) = x^3 + x^2 - x + 9, we can use the power rule. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). Applying this rule, we have:

f'(x) = 3x^2 + 2x - 1

To find the critical value of f(x) on the interval [-2, 0], we need to find the values of x where f'(x) = 0 or f'(x) is undefined. Setting f'(x) = 0 and solving for x, we get:

3x^2 + 2x - 1 = 0

This quadratic equation can be factored as (3x - 1)(x + 1) = 0. So the critical values of f(x) on the interval [-2, 0] are x = -1 and x = 1/3.